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While early authorities are pretty universally consistent about the current applicability of this rule ("in all times and in all places"), there are three main explanations offered in relatively later works to explain the common lax attitude about this Mitzva:

Bakh: The prohibition only applies to gentiles' Chodesh. In other words, there is a prohibition to celebrate things like New Year's Day, April Fools' Day, May Day, and Canada Day, but not Rosh Chodesh of the Jewish months. While indeed sometimes the Jewish Rosh Chodesh falls on the first of a secular month, we need not be concerned for this minority of cases. (It's worth noting this is quite the novel position, as according to nearly all prior opinions the secular date is irrelevant to Halakha.)

Ramah: We apply a Sefeik Sefeika (double doubt): maybe today isn't really the first of the month because everything is shifted by a few quarter days before a leap year, and even if not maybe the Gregorian calendar is wrong and the first of the month was 11 days ago. With so many doubts we need not be concerned about this prohibition. (The Shach there disputes this as being only one doubt: is it Rosh Chodesh or not?)

TAS / Aroch Hashulkhan: The prohibition doesn't apply far away from Israel. God knew that in lands far from Israel they wouldn't be able to know when Rosh Chodesh would be declared and the prohibition would be too hard to keep, so He never enacted the prohibition in lands far from Israel. This confusion is evident in the modern practice of sometimes celebrating 2 days of Rosh Chodesh out of doubt. (Some question the application of this to our times when there is no Sha'at HaDechak as we can all readily find out certified Rosh Chodesh times.)

Indeed while all of those opinions have holes in them, we can ask a stronger question: Numbers 10:10 seems to list Rosh Chodesh among the happy days of the year, in that, in contrast to Numbers 10:9, the Korbanot in the Mikdash were accompanied by Teki'ah blasts instead of Teru'ah blasts. How then could celebration on Rosh Chodesh be prohibited?

The answer though is clear. The real prohibition of Chodesh only applied to women, as we see from Numbers 11:20 which limits the negative effects of celebrating Rosh Chodesh to that which comes from men's noses (not ribs). Since women don't participate in the offerring of sacrifices in the Temple (Kiddushin 36a), there is no question from the celebratory procedures there on Rosh Chodesh.

It's important to note that for those obligated to celebrate Rosh Chodesh, fasting would be prohibited (Ta'anit 2:10, ShA OC 418:1). However, while eating on this holiday, they should be careful to seek foods certified Kosher with Yoshon flour, as it says (Vayikra 26:10) "In the face of the [Rosh] Chodesh, eat only Yoshon."

חדש אסור מן התורה means that it is forbidden to celebrate months that are from the Torah.

As we know well, the months in the Torah do not have names. They only have numbers. Yet when we declare an upcoming new month at birkat haHodesh the month is given a name. Of course, this name is the name of a Canaanite month. This is the month that we are celebrating on Rosh Hodesh. It's a neat little trick that allows us to fulfill the mitzvah of observing Rosh Hodesh while still not accidentally commemorating a month of Torah origin.

The prohibition was only to celebrate Rosh Chodesh in a mundane environment. But, with the advent of the Davidic Dynasty and the emminent construction of the Holy Temple, Rosh Chodesh became a prevalent celebration.